Problem 10 - Precalculus

Question

Suppose (2,-3) is on the graph of \(y=f(x) .\) In Exercises \(1-18,\) use Theorem 1.7 to find a point on the graph of the given transformed function. $$ y=2 f(x+1) $$

Step-by-Step Solution

Answer

The point on the graph of the transformed function is (1, -6).

Step 1: Understanding the Transformation

The given transformation is \( y = 2f(x+1) \). This involves two transformations of \( y = f(x) \): a horizontal shift and a vertical stretch.

Step 2: Horizontal Shift

The function \( y = f(x+1) \) indicates a shift to the left by 1 unit. If the original point \((x, y)\) is \((2, -3)\), then the new \( x \)-coordinate will be \( x - 1 = 2 - 1 = 1 \).

Step 3: Vertical Stretch

The function \( y = 2f(x+1) \) indicates that the output of \( f(x+1) \) is multiplied by 2. If the original \( y \)-value at \( x+1 = 2 \) (or \( x = 1 \)) is \(-3\), then the new \( y \)-value will be \( 2 \times (-3) = -6 \).

Step 4: Calculate the New Point

Combining both transformations, the new point on the graph \( y = 2f(x+1) \) is \((1, -6)\).

Key Concepts

Horizontal Shift

One common transformation in functions is the horizontal shift, which affects the position of the graph along the x-axis. To achieve a horizontal shift, we modify the function's input value (x). In the transformation \( y = f(x + 1) \), the function is shifted to the left by one unit. This adjustment is always counterintuitive for many beginners.

- If you add to \( x \), the graph shifts left.- If you subtract from \( x \), the graph shifts right.

In our problem, the given point \((2, -3)\) moves to \( (2 - 1, -3) = (1, -3) \) after the horizontal shift. The x-coordinate decreases since we are subtracting one, signifying a leftward movement on the graph. It's essential to grasp this concept, as it applies to all function types, not just linear ones.

Vertical Stretch

Another transformation that we encounter is the vertical stretch. This affects how "tall" or "short" a graph looks. For the transformation \( y = 2f(x+1) \), the term 2 is multiplying the output of the function, thus causing a vertical stretch.

- A multiplication factor greater than 1 stretches the graph vertically.- A factor between 0 and 1 compresses it vertically.

In this example, the original y-value is \(-3\) at \( x = 1 \) (after the horizontal shift). When we multiply this by 2, we get \( 2 \times (-3) = -6 \). This means that every point on the graph is twice as far from the x-axis compared to the original function, resulting in a more "stretched" appearance.

Precalculus

Precalculus involves a lot of groundwork preparing students for calculus. In the realm of precalculus, understanding function transformations is crucial, as it sets the foundation for working with more advanced mathematical concepts.

Key topics like horizontal shifts and vertical stretches are just a part of what you'll encounter in precalculus. You also dive into:

• Polynomials
• Trigonometry
• Exponential functions
• Logarithms

Functional transformations, such as those discussed here, should be seen as building blocks for understanding how equations represent movements and changes. Mastery of these concepts now will make the leap into calculus smoother, as you'll already understand how various transformations affect graphs and functions. Always approach the studies in this area with patience and practice, as these are the skills that will carry forward into more complex calculus topics.